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One can specify the following mathematical characteristics in each of the five Platonic solids:

1. The radius of the sphere circumscribing the polyhedron;

2. The radius of the sphere inscribed in the polyhedron;

3. The surface area of the polyhedron;

4. The volume of the polyhedron.

Tetrahedron: All four faces are equilateral triangles.

The radius of a circumscribed sphere

of tetrahedron is

where a is side length.

The radius of an inscribed sphere

of tetrahedron is

The surface area of tetrahedron

It is possible to represent the surface area of tetrahedron as an area of the model for better illustration.

The volume of tetrahedron is

Octahedron: All eight faces are equilateral triangles.

The radius of a circumscribed sphere

of an octahedron is

where a is side length.

The radius of an inscribed sphere

of octahedron is

The surface area of octahedron is

It is possible to represent the surface area of octahedron as an area of the model for better illustration.

The volume of octahedron is

Hexahedron (cube): All six faces are squares.

The radius of a circumscribed sphere

of cube

where a is side length.

The radius of an inscribed sphere

of cube is

The surface area of cube is

It is possible to represent the surface area of cube as an area of the model for better illustration.

The volume of cube is

Dodecahedron: All 12 faces are regular pentagons.

The radius of a circumscribed sphere

of dodecahedron

where a is side length.

The radius of an inscribed sphere

of dodecahedron is

The surface area of dodecahedron is

It is possible to represent the surface area of dodecahedron as an area of the model for better illustration.

The volume of dodecahedron is

Icosahedron: All 20 faces are equilateral triangles.

The radius of a circumscribed sphere

of icosahedron

where a is side length.

The radius of an inscribed sphere

of icosahedron is

The surface area of icosahedron is

It is possible to represent the surface area of icosahedron as an area of the model for better illustration.

the volume of icosahedron is

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